Severi curves of rational elliptic surfaces

2023/06/05 by François Greer, Greer, François, Joseph Helfer +3
Computer Science · Engineering · Mathematics · #14J27 (Primary) 14N10 (Secondary) #Advanced Numerical Analysis Techniques #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2306.03200

openalex publication_date 2023/06/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study Severi curves parametrizing rational bisections of elliptic fibrations associated to general pencils of plane cubics. Our main results show that these Severi curves are connected and reduced, and we give an upper bound on their geometric genus using quasi-modular forms. We conjecture that these Severi curves are eventually reducible, and we formulate a precise conjecture for their degrees in ℙ², featuring a divisor sum formula for collision multiplicities of branch points.

Related

• Counting curves of any genus on rational ruled surfaces
• Negative Curves and Elliptic Fibrations on a Special Rational Surface
• Severi Varieties on Ruled Surfaces over Elliptic Curves
• Severi varieties on Enriques surfaces of base change type and on rational elliptic surfaces
• Linear sections of the Severi variety and moduli of curves